Proof Definition and 999 Threads

A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in all possible cases. An unproven proposition that is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work.Proofs employ logic expressed in mathematical symbols, along with natural language which usually admits some ambiguity. In most mathematical literature, proofs are written in terms of rigorous informal logic. Purely formal proofs, written fully in symbolic language without the involvement of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics, oral traditions in the mainstream mathematical community or in other cultures. The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.

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  1. D

    Can the Limit of a Function Exist Despite Contradictory Values?

    Homework Statement Proof that the limit of the function below doesn't exists. limx-->1 1/(x-1)Homework EquationsThe Attempt at a Solution Lets assume that limit L exists. So if (1) 0< |x-1| < δ then (2) |1/(x-1) - L| < ε at the book they gave an example by giving a value...
  2. M

    MHB Proof of Sets: Proving (i) and (ii)

    If $X$ is a set, then the power set $P(X)$ of a set is the set of all subsets of $X$. I need to decide whether the following statements are true or false and prove it: (i) If $Z = X \cup Y$ , then $P(Z) = P(X) \cup P(Y)$. (ii) If $Z = X \cap Y$ , then $P(Z) = P(X) \cap P(Y)$. By examples I...
  3. M

    MHB Proof of Parallelogram ABCD: Midpoint X & Y Show Area $\frac{1}{4}$

    ABCD is a parallelogram . X is the midpoint of AD & Y is the midpoint of BC. Show that the area of $\triangle {ABX}$ is $\frac{1}{4}$ the area of ABCD Can you help me with this proof ? were should i start ? I think It should be by proving $\triangle{DBC} \cong \triangle{DBA} $ using SAS as...
  4. JulienB

    Proof of differentiability for <x,x>

    Homework Statement Hi everybody! I'm struggling to solve the following problem: Let ##< \cdot, \cdot >## be an inner product on the vector space ##X##, and ##|| \cdot ||## is the norm generated by the inner product. Prove that the function ##x \in X \mapsto ||x||^2 \in \mathbb{R}## is...
  5. A

    Solve Set Proof Problems Homework Statement

    Homework Statement Can anyone please help me solve these questions? (1) Prove that (A-B) - (B-C) = A-B (2)Simplify (A-( A N B)) N (B-(ANB)) (3) Simplify ( ( A N ( B U C)) N ( A-B)) N ( B U C') (4)Use element property and algebraic argument to derive the property (A-B) U (B-C) = (A U B) - (B N...
  6. T

    I Motivation and proof behind cross products

    this question is a repost from math stackexchange because that guy worded the question so perfectly the question i really wanted to ask about cross products. *please see image below* as far i can understand, the formula for the cross product is basically that the idea of a cross product is sort...
  7. A

    Prove by Induction: $w_k = w_{k-2} + k$

    Homework Statement Prove by induction $$w_k = w_{k−2} + k$$, for all integers $$k \ge 3, w_1 = 1,w_2 = 2$$ has an explicit formula $$ w_n =\begin{cases} \frac{(n+1)^2}{4}, & \text{if $n$ is odd} \\ \frac n2(\frac n2 + 1), & \text{if $n$ is even} \end{cases}$$ Homework Equations The Attempt...
  8. T

    I Are There n!/2 Even/Odd Permutation Matrices for nxn?

    is there a proof that the number of even/odd permutation matrices of any nxn, where n is greater than 3, is n!/2? basically, i want to understand the derivation of n!/2. thank you!
  9. P

    MHB Is Cantor's second diagonal proof valid?

    Cantor "proved" that if there was a list that purported to include all irrational numbers, then he could find an irrational number that was not on the list. Please consider two scenarios: 1. The list claims to contain all irrationals but doesn't. 2. The list absolutely contains all...
  10. alexmahone

    MHB How do I complete this convergence proof?

    Prove that if a subsequence of a Cauchy sequence converges then so does the original Cauchy sequence. I'm assuming that we're not allowed to use the fact that every Cauchy sequence converges. Here's my attempt: Let $\displaystyle\{s_n\}$ be the original Cauchy sequence. Let $\displaystyle...
  11. M

    I Sum principle proof: discrete mathematics

    Theorem: Let ##A_1, A_2, ..., A_k## be finite, disjunct sets. Then ##|A_1 \cup A_2 \cup \dots \cup A_k| = |A_1| + |A_2| + \dots + |A_k|## I will give the proof my book provides, I don't understand several parts of it. Proof: We have bijections ##f_i: [n_i] \rightarrow A_i## for ##i \in [k]##...
  12. Y

    MHB Formal proof using the deduction theorem

    Hello everyone, I am trying to find a proof for: \[\vdash \left ( \sim \alpha \rightarrow \sim \left ( \sim \alpha \right ) \right )\rightarrow \alpha\] I am using the L inference system, which includes the modus ponens inference rule, and the axioms and statements attached below. That's the...
  13. A

    I Proof to the Expression of Poisson Distribution

    Hello. Given a range of time in which an event can occur an indefinite number of times, we say a random variable X folows a poisson distribution when it follows this statements: X is the number of times an event occurs in an interval and X can take values 0, 1, 2, … The occurrence of one event...
  14. I

    Discrete Math Proof: Necessary Condition for Divisibility by 6

    Homework Statement We have JUST started writing proofs recently, and I am a little bit doubtful in my abilities in doing this, so I just want to verify that my proof actually works. I was expecting this one to be a lot longer since the previous 2 were. I don't see any glaring flaws in it, but...
  15. M

    I Proof that every basis has the same cardinality

    Hello all. I have a question concerning following proof, Lemma 1. http://planetmath.org/allbasesforavectorspacehavethesamecardinalitySo, we suppose that A and B are finite and then we construct a new basis ##B_1## for V by removing an element. So they choose ##a_1 \in A## and add it to...
  16. N

    I How does Bell make this step in his proof?

    From drchinese website http://www.drchinese.com/David/Bell_Compact.pdf on page 406 there are 2 equations at the top. How do you get from the top one to the second one? There is a hint about using (1) but I think it cannot be done. You might be able to do it with other assumptions but I think...
  17. kubaanglin

    Projectile motion equation proof

    Homework Statement Show that the launch angle θ is given by the expression: θ=tan-1(4hmax/R) where hmax is the maximum height in the trajectory and R is the range of the projectile. Homework Equations hmax=vi2sin2(θ)/2g R=vi2sin(2θ)/g The Attempt at a Solution I am trying to understand the...
  18. M

    Proof of A Union of A Intersection B Equals A

    Homework Statement Prove that ##A \cup (A \cap B) = A## Homework Equations In the previous exercise, we proved: Let A, B be sets. Then, the following statements are equivalent: 1) ##A \subseteq B## 2) ##A \cup B = B## 3) ##A \cap B = A## The Attempt at a Solution The proof of ##A \cup (A...
  19. weezy

    Proof of independence of position and velocity

    A particle's position is given by $$r_i=r_i(q_1,q_2,...,q_n,t)$$ So velocity: $$v_i=\frac{dr_i}{dt} = \sum_k \frac{\partial r_i}{\partial q_k}\dot q_k + \frac{\partial r_i}{\partial t} $$ In my book it's given $$\frac{\partial v_i}{\partial \dot q_k} = \frac{\partial r_i}{\partial q_k}$$...
  20. K

    Is the Function f(x) = x^2 Injective?

    Homework Statement Prove that ##f: \mathbb{R}\to\mathbb{R}, f(x) = x^2## is not injective. Homework Equations Definition of an injection: function ##f:A\to B## is an injection if and only if ##\forall a,b \in A, f(a) = f(b) \Rightarrow a = b##. The Attempt at a Solution ##f...
  21. H

    I Proof of convergence & divergence of increasing sequence

    I'm using the book of Jerome Keisler: Elementary calculus an infinitesimal approach. I have trouble understanding the proof of the following theorem. I'm not sure what it means. Theorem: "An increasing sequence <Sn> either converges or diverges to infinity." Proof: Let T be the set of all real...
  22. weezy

    Verifying the Correctness of My Proof

    1. I have to show: 2. Given: 3. My attempt : I just want to verify if what I've done is correct or not. Thanks!
  23. e2m2a

    A Circular reasoning and proof by Contradiction

    I need to understand something about proof by contradiction. Suppose there is an expression "a" and it is known to be equal to expression "b". Furthermore, suppose it is conjectured that expression "c" is also equal to expression "a". This would imply expression "c" is equal to expression...
  24. G

    A Steps in proof for Eotvos' law

    I have purchased an article after recommendation on wikipedia that as far as I am aware proves eotvos law. Here is a quote from wikipedia from this site: https://en.wikipedia.org/wiki/Eötvös_rule: ''John Lennard-Jones and Corner published (1940) a derivation of the equation by means of...
  25. Battlemage!

    Is this a valid proof for n >2^n for all n>3

    Homework Statement Show that n!>2n for all n>3. Homework Equations I will attempt to use induction. The Attempt at a Solution We want to show that n!>2n for all n>3. Consider the case when n=4. 4! = 24 > 2^4 =16. We want to show by way of induction that if the inequality is true for...
  26. PhotonSSBM

    Boolean Algebra Proof (Distribution and XOR)

    Homework Statement Use the definition of exclusive or (XOR), the facts that XOR commutes and associates (if you need this) and all the non-XOR axioms and theorems you know from Boolean algebra to prove this distributive rule: A*(B (XOR) C) = (A*B) (XOR) (A*C) Homework Equations All the...
  27. D

    I Confusion on Bianchi Identity proof

    This is from a general relativity book but I think this is the appropriate location. The proof that \nabla_{[a} {R_{bc]d}}^e=0 is as follows: Choose coordinates such that \Gamma^a_{bc}=0 at an event. We have \nabla_a {R_{bcd}}^e = \partial_a \partial_b \Gamma^e_{cd} - \partial_a...
  28. dumbdumNotSmart

    Complex Conjugate Inequality Proof

    Homework Statement $$ \left | \frac{z}{\left | z \right |} + \frac{w}{\left | w \right |} \right |\left ( \left | z \right | +\left | w \right | \right )\leq 2\left | z+w \right | $$ Where z and w are complex numbers not equal to zero. 2.$$\frac{z}{\left | z \right | ^{2}}=\frac{1}{\bar{z}}$$...
  29. M

    I Recursion theorem: application in proof

    I have read a proof but I have a question. To give some context, I first wrote down this proof as written in the book. First, I provide the recursion theorem though. Recursion theorem: Let H be a set. Let ##e \in H##. Let ##k: \mathbb{N} \rightarrow H## be a function. Then there exists a...
  30. J

    MHB Proof: polynomial with integer solutions

    I am stuck with one proof and I need some help because I don't have any idea how to proceed at this moment. The task says: If f(x) is a polynomial with integer coefficients, and if f(a)=f(b)=f(c)=-1, where a,b,c are three unequal integers, the equation f(x)=0 does not have integer solutions...
  31. Priyadarshini

    B What is the error in this proof attempting to show 0/0 = 2?

    Recently I can across this proof: 0/0= (100-100)/(100-100) = (10^2-10^2)/10(10-10) =(10-10)(10+10)/10(10-10) = (10+10)/10 =20/10 = 2 But this is obviously wrong, as 0/0 is infinity, but which line in this proof is actually wrong?
  32. G

    Looking for proof of Superpositional Energy Conservation

    Let's have waves with their carried power proportional to the square of their amplitude(s). The waves obey the principle of superposition. Before superposition, we can calculate the power output based on the amplitudes. After superposition, there will be new values for the amplitudes, but the...
  33. Austin Chang

    Function Injectivity Subjectivity Proof

    1. Homework Statement Proof Let F:A→B and g: B→C be functions. Suppose that g°f is injective. Prove that f is injective. Homework EquationsThe Attempt at a Solution Let x,y ∈ A, and suppose g°f (x) = g°f(y) and x = y. Suppose g: B→C was not injective. then f(g(x)), if g(x) is some element...
  34. M

    I Is the negation of the statement in the book correct too?

    Hello. I'm currently studying about natural numbers and I encountered the theorem of definition by recursion: This states: Let ##H## be a set, let ##e \in H## and let ##k: H \rightarrow H## be a function. Then there is a unique function ##f: \mathbb{N} \rightarrow H## such that ##f(1) = e##...
  35. Powergade

    Proof that time <0 for tachyons > c

    1. Homework Statement Derive the result DeltaT <0 for U> (sqrt(1-v^2/c^2)+1)/v/c)c Homework Equations DeltaT = u/l + (u-v/1-uv/c^2)1/l Where: DeltaT is the time for the tachyon to go and come back. u is the velocity of the tachyon l is the distance that the tachyon goes v is the velocity...
  36. nmsurobert

    How do I prove Lorentz Invariance using 4-vectors?

    Homework Statement I'm asked to prove that Et - p⋅r = E't' - p'⋅r' Homework Equations t = γ (t' + ux') x = γ (x' + ut') y = y' z = z' E = γ (E' + up'x) px = γ (p'x + uE') py = p'y pz = p'z The Attempt at a Solution Im still trying to figure out 4 vectors. I get close to the solution but I...
  37. S

    I Spivak - Proof of f(x) = c on [a, b]

    In Spivak's Calculus, on page 121 there is this theorem Then he generalizes that theorem: I tried proving theorem 4 on my own, before looking at Spivak's proof. Thus I let c = 0 and then by theorem 1, my proof would be completed. Is this a correct proof? Spivak's proof for theorem 4...
  38. S

    B Proof of a lemma of BÉZOUT’S THEOREM

    Hi, One silly thing is bothering me. As per one lemma, If a, b, and c are positive integers such that gcd(a, b) = 1 and a | bc, then a | c. This is intuitively obvious. i.e. Since GCD is 1 'a' does not divide 'b'. Now, 'a' divides 'bc' so, 'a' divides 'c'. Proved. What is bothering me is ...
  39. S

    Proof involving central acceleration and vector products

    Homework Statement Suppose r:R\rightarrow { V }_{ 3 } is a twice-differentiable curve with central acceleration, that is, \ddot { r } is parallel with r. a. Prove N=r\times \dot { r } is constant b. Assuming N\neq 0, prove that r lies in the plane through the origin with normal N. Homework...
  40. M

    I Proving the Implication of p and (p -> q) to q without Truth Tables

    Hello everyone! I want to proof that: ##p \land (p \to q) \Rightarrow q## I know this is a quite trivial problem using truth tables, however, I want to do it without it. As I'm learning this myself, is this the correct approach? ##p \land (p \to q)## ##\iff p\land (\neg p \lor q)## ##\iff (p...
  41. T

    B Some trouble understanding this basic inequality proof

    just starting out with proofs... i tried manipulating the right side of the inequality, but i don't see why it's equal to (n+1)!
  42. J

    I Proof Using Rearrangement Inequality

    The Rearrangement Inequality states that for two sequences ##{a_i}## and ##{b_i}##, the sum ##S_n = \sum_{i=1}^n a_ib_i## is maximized if ##a_i## and ##b_i## are similarly arranged. That is, big numbers are paired with big numbers and small numbers are paired with small numbers. The question...
  43. S

    Prove this is a right triangle in a sphere

    Homework Statement Let P be a point on the sphere with center O, the origin, diameter AB, and radius r. Prove the triangle APB is a right triangle Homework Equations |AB|^2 = |AP|^2 + |PB|^2 |AB}^2 = 4r^2 The Attempt at a Solution Not sure if showing the above equations are true is the...
  44. TheSodesa

    Proving two simple matrix product properties

    Homework Statement Let ##A## be an n × p matrix and ##B## be an p × m matrix with the following column vector representation, B = \begin{bmatrix} b_1 , & b_2, & ... & ,b_m \end{bmatrix} Prove that AB = \begin{bmatrix} Ab_1 , & Ab_2, & ... & , Ab_m \end{bmatrix} If ##A## is represented...
  45. binbagsss

    QM Bra & Ket Linear Algebra Hermitian operator proof -- quick question

    Homework Statement Hi, Just watching Susskind's quantum mechanics lecture notes, I have a couple of questions from his third lecture: Homework Equations [/B] 1) At 25:20 he says that ## <A|\hat{H}|A>=<A|\hat{H}|A>^*## [1] ##<=>## ##<B|\hat{H}|A>=<A|\hat{H}|B>^*=## [2] where ##A## and ##B##...
  46. M

    Understanding the Position Vector in Calculus Problems

    Homework Statement With ##\vec{r}## the position vector and ##r## its norm, we define $$ \vec{f} = \frac{\vec{r}}{r^n}.$$ Show that $$ \nabla^2\vec{f} = n(n-3)\frac{\vec{r}}{r^{n+2}}.$$ Homework Equations Basic rules of calculus. The Attempt at a Solution From the definition of...
  47. M

    MHB Proof That Radius of Melting Snowball Decreases Constantly

    A spherical snowball is melting at a rate proportional to its surface area. That is, the rate at which its volume is decreasing at any instant is proportional to its surface area at that instant. (i) Prove that the radius of the snowball is decreasing at a constant rate. can someone help me?
  48. D

    Linear Algebra with Proof by Contradiction

    This is a linear algebra question which I am confused. 1. Homework Statement Prove that "if the union of two subspaces of ##V## is a subspace of ##V##, then one of the subspaces is contained in the other". The Attempt at a Solution Suppose ##U##, ##W## are subspaces of ##V##. ##U \cup W##...
  49. D

    Contradiction Method: Proving Statements Through Contradiction and Supposition

    Proof by contradiction starts by supposing a statement, and then shows the contradiction. 1. Homework Statement Now, there is a statement ##A##. Suppose ##A## is false. It leads to contradiction. So ##A## is true. My question: There are two statements ##A## and ##B##. Suppose ##A## is true...
  50. A

    How to Prove This Hermitian Operator Statement?

    1. Homework Statement prove the following statement: Hello, can someone help me prove this statement A is hermitian and {|Ψi>} is a full set of functions Homework Equations Σ<r|A|s> <s|B|c>[/B]The Attempt at a Solution Since the right term of the equation reminds of the standard deviation, I...
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